1 Inference About a Population Variance Sometimes we are interested in making inference about the variability of processes. Examples: –Investors use variance as a measure of risk. To draw inference about variability, the parameter of interest is  2.

2 The sample variance s 2 is an unbiased, consistent and efficient point estimator for  2. The statistic has a distribution called Chi- squared, if the population is normally distributed. d.f. = 5 d.f. = 10 Inference About a Population Variance

3 Testing and Estimating a Population Variance From the following probability statement P(  2 1-  /2 <  2 <  2  /2 ) = 1-  we have (by substituting  2 = [(n - 1)s 2 ]/  2.)

4 Example –H 0 :  2 = 1 H 1 :  2 <1 Testing the Population Variance

5 Inference about Two Populations Variety of techniques are presented whose objective is to compare two populations. We are interested in: –The difference between two means. –The ratio of two variances.

6 Two random samples are drawn from the two populations of interest. Because we compare two population means, we use the statistic. 13.2Inference about the Difference between Two Means: Independent Samples

7 1. is normally distributed if the (original) population distributions are normal. 2. is approximately normally distributed if the (original) population is not normal, but the samples’ size is sufficiently large (greater than 30). 3. The expected value of is  1 -  2 4. The variance of is  1 2 / n 1 +  2 2 / n 2 The Sampling Distribution of

8 If the sampling distribution of is normal or approximately normal we can write: Z can be used to build a test statistic or a confidence interval for  1 -  2 Making an inference about    –  

9 Practically, the “Z” statistic is hardly used, because the population variances are not known. ? ? Instead, we construct a t statistic using the sample “variances” (S 1 2 and S 2 2 ). S22S22 S12S12 t Making an inference about    –  

10 Two cases are considered when producing the t-statistic. –The two unknown population variances are equal. –The two unknown population variances are not equal. Making an inference about    –  

11 Inference about    –   : Equal variances Example: s 1 2 = 25; s 2 2 = 30; n 1 = 10; n 2 = 15. Then, Calculate the pooled variance estimate by: n 2 = 15 n 1 = 10 The pooled Variance estimator

12 Inference about    –   : Equal variances Construct the t-statistic as follows: Perform a hypothesis test H 0 :     = 0 H 1 :     > 0 or < 0or 0 Build a confidence interval

13 Inference about    –   : Unequal variances

14 Inference about    –   : Unequal variances Conduct a hypothesis test as needed, or, build a confidence interval

15 Inference about the ratio of two variances In this section we draw inference about the ratio of two population variances. This question is interesting because: –Variances can be used to evaluate the consistency of processes. –The relationship between population variances determines which of the equal-variances or unequal- variances t-test and estimator of the difference between means should be applied

16 Parameter to be tested is  1 2 /  2 2 Statistic used is Parameter and Statistic Sampling distribution of the statistic The statistic [ s 1 2 /  1 2 ] / [ s 2 2 /  2 2 ] follows the F distribution with 1 = n 1 – 1, and 2 = n 2 – 1.

17 –Our null hypothesis is always H 0 :  1 2 /  2 2 = 1 –Under this null hypothesis the F statistic becomes F = S12/12S12/12 S22/22S22/22 Parameter and Statistic